{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# RT0-P0 Element for Poisson Equation in 2D"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This example is to show the rate of convergence of the linear finite element approximation of the Poisson equation on the unit square:\n",
    "\n",
    "$$- \\nabla (d \\nabla) u = f \\; \\hbox{in } (0,1)^2$$\n",
    "\n",
    "for the following boundary conditions\n",
    "- pure Dirichlet boundary condition: $u = g_D \\text{ on } \\partial \\Omega$.\n",
    "- Pure Neumann boundary condition: $d\\nabla u\\cdot n=g_N \\text{ on } \\partial \\Omega$.\n",
    "- mixed boundary condition: $u=g_D \\text{ on }\\Gamma_D, \\nabla u\\cdot n=g_N \\text{ on }\\Gamma_N.$\n",
    "\n",
    "Find $(\\sigma , u)$ in $H_{g_N,\\Gamma_N}(div,\\Omega)\\times L^2(\\Omega)$ s.t. \n",
    "\n",
    "$$ (d^{-1}\\sigma,\\tau) + (div \\tau, u)  = \\langle \\tau \\cdot n, g_D \\rangle_{\\Gamma_D} \\quad \\forall \\tau \\in H_{0,\\Gamma_N}(div,\\Omega)$$\n",
    "\n",
    "$$ (div \\sigma, v)                =  -(f,v) \\quad \\forall v \\in L^2(\\Omega) $$\n",
    " \n",
    " where \n",
    " \n",
    " $$H_{g,\\Gamma}(div,\\Omega) = \\{\\sigma \\in H(div,\\Omega); \\sigma \\cdot n = g  \\text{ on } \\Gamma \\subset \\partial\\Omega \\}.$$\n",
    "\n",
    " The unknown $sigma = d\\nabla u$ is approximated using the lowest order\n",
    " Raviart-Thomas element (RT0) and $u$ by piecewise constant element (P0).\n",
    "\n",
    "**References**\n",
    "\n",
    "\n",
    "**Subroutines**:\n",
    "\n",
    "    - PoissonRT0\n",
    "    - squarePoissonRT0\n",
    "    - mfemPoisson\n",
    "    - PoissonRT0mfemrate\n",
    "    \n",
    "The method is implemented in `PoissonRT0` subroutine and tested in `squarePoissonRT0`. Together with other elements (BDM1), `mfemPoisson` provides a concise interface to solve Poisson equation in mixed formulation. The RT0-P0 element is tested in `PoissonRT0mfemrate`. This doc is based on `PoissonRT0mfemrate`.    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## P1 Linear Element\n",
    "\n",
    "For the linear element on a simplex, the local basis functions are\n",
    "barycentric coordinate of vertices. The local to global pointer is\n",
    "`elem`. This is the simplest and default element for elliptic equations.\n",
    "\n",
    "**A local basis of P1**\n",
    "\n",
    "For $i = 1, 2, 3$, a local basis of the linear element space is given by the barycentric coordinate\n",
    "\n",
    "$$\\phi_i = \\lambda_i, \\quad \\nabla \\phi_i = \\nabla \\lambda_i = - \\frac{|e_i|}{2|T|}\\boldsymbol n_i, \\quad \\int_T \\nabla \\phi_i\\nabla \\phi_j = \\frac{1}{4|T|}\\boldsymbol l_i \\cdot \\boldsymbol l_j$$\n",
    "\n",
    "where $e_i$ is the edge opposite to the i-th vertex and $\\boldsymbol n_i$ is the unit\n",
    "outwards normal direction, and $\\boldsymbol l_i$ is the edge vector of $e_i$.  \n",
    "\n",
    "See [Finite Element Methods](http://www.math.uci.edu/~chenlong/226/Ch2FEM.pdf) Section 2.1 for geometric explanation of the barycentric coordinate and [Programming of Finite Element Methods in MATLAB](http://www.math.uci.edu/~chenlong/226/Ch3FEMcode.pdf) for detailed explanation. For P1 element, the basic data structure `node,elem` is sufficient and displayed for the following coarse mesh."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "elem =\n",
      "\n",
      "     8     6     2\n",
      "     7     6     4\n",
      "     5     6     1\n",
      "     9     6     3\n",
      "     8     3     6\n",
      "     7     1     6\n",
      "     5     2     6\n",
      "     9     4     6\n",
      "\n"
     ]
    },
    {
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"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "imatlab_export_fig('print-jpeg')\n",
    "node = [0,0; 1,0; 1,1; 0,1];\n",
    "elem = [2,3,1; 4,1,3];      \n",
    "[node,elem] = uniformbisect(node,elem);\n",
    "figure('rend','painters','pos',[10 10 225 225])\n",
    "showmesh(node,elem);\n",
    "findnode(node);\n",
    "findelem(node,elem);\n",
    "display(elem);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/jpeg": 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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "% Setting\n",
    "[node,elem] = squaremesh([0,1,0,1],0.25); \n",
    "mesh = struct('node',node,'elem',elem);\n",
    "figure('rend','painters','pos',[10 10 225 225])\n",
    "showmesh(node,elem);\n",
    "option.L0 = 3;\n",
    "option.maxIt = 4;\n",
    "option.printlevel = 1;"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Mixed boundary condition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Warning: File: /Dropbox/Math/Programming/ifem/solver/mg.m Line: 695 Column: 24\n",
      "Defining \"Di\" in the nested function shares it with the parent function.  In a future release, to share \"Di\" between parent and nested functions, explicitly define it in the parent function.\n",
      "> In Poisson (line 233)\n",
      "  In femPoisson (line 65)\n",
      "Multigrid V-cycle Preconditioner with Conjugate Gradient Method\n",
      "#dof:     4225,  #nnz:    19906, smoothing: (1,1), iter: 10,   err = 1.55e-09,   time = 0.11 s\n",
      "Multigrid V-cycle Preconditioner with Conjugate Gradient Method\n",
      "#dof:    16641,  #nnz:    80770, smoothing: (1,1), iter: 10,   err = 1.55e-09,   time = 0.096 s\n",
      "Multigrid V-cycle Preconditioner with Conjugate Gradient Method\n",
      "#dof:    66049,  #nnz:   325378, smoothing: (1,1), iter: 10,   err = 1.47e-09,   time =  0.2 s\n",
      "\n",
      " #Dof       h       ||u-u_h||    ||Du-Du_h||   ||DuI-Du_h|| ||uI-u_h||_{max}\n",
      "\n",
      " 1089   3.12e-02   1.15027e-03   1.08974e-01   2.21506e-03   9.04547e-04\n",
      " 4225   1.56e-02   2.88013e-04   5.45135e-02   5.54571e-04   2.26928e-04\n",
      "16641   7.81e-03   7.20310e-05   2.72601e-02   1.38693e-04   5.67600e-05\n",
      "66049   3.91e-03   1.80095e-05   1.36305e-02   3.46767e-05   1.41918e-05\n",
      "\n",
      " #Dof   Assemble     Solve      Error      Mesh    \n",
      "\n",
      " 1089   7.00e-02   1.36e-02   9.00e-02   1.00e-02\n",
      " 4225   3.00e-02   1.09e-01   4.00e-02   2.00e-02\n",
      "16641   1.10e-01   9.62e-02   8.00e-02   6.00e-02\n",
      "66049   4.00e-01   2.00e-01   2.30e-01   2.40e-01\n",
      "\n",
      "\n"
     ]
    },
    {
     "data": {
      "image/jpeg": 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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "option.plotflag = 0;\n",
    "pde = sincosdata;\n",
    "mesh.bdFlag = setboundary(node,elem,'Dirichlet','~(x==0)','Neumann','x==0');\n",
    "femPoisson(mesh,pde,option);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Pure Neumann boundary condition\n",
    "\n",
    "When pure Neumann boundary condition is posed, i.e., $-\\Delta u =f$ in $\\Omega$ and $\\nabla u\\cdot n=g_N$ on $\\partial \\Omega$, the data should be consisitent in the sense that $\\int_{\\Omega} f \\, dx + \\int_{\\partial \\Omega} g \\, ds = 0$. The solution is unique up to a constant. A post-process is applied such that the constraint $\\int_{\\Omega}u_h dx = 0$ is imposed. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Multigrid V-cycle Preconditioner with Conjugate Gradient Method\n",
      "#dof:     4225,  #nnz:    20860, smoothing: (1,1), iter: 12,   err = 1.43e-09,   time = 0.03 s\n",
      "Multigrid V-cycle Preconditioner with Conjugate Gradient Method\n",
      "#dof:    16641,  #nnz:    82684, smoothing: (1,1), iter: 12,   err = 6.33e-09,   time = 0.08 s\n",
      "Multigrid V-cycle Preconditioner with Conjugate Gradient Method\n",
      "#dof:    66049,  #nnz:   329212, smoothing: (1,1), iter: 13,   err = 2.40e-09,   time = 0.26 s\n",
      "\n",
      " #Dof       h       ||u-u_h||    ||Du-Du_h||   ||DuI-Du_h|| ||uI-u_h||_{max}\n",
      "\n",
      " 1089   3.12e-02   1.29973e-03   1.08855e-01   5.80361e-03   3.86104e-03\n",
      " 4225   1.56e-02   3.25931e-04   5.44960e-02   1.64867e-03   1.14414e-03\n",
      "16641   7.81e-03   8.15520e-05   2.72576e-02   4.64515e-04   3.30465e-04\n",
      "66049   3.91e-03   2.03927e-05   1.36301e-02   1.29765e-04   9.37017e-05\n",
      "\n",
      " #Dof   Assemble     Solve      Error      Mesh    \n",
      "\n",
      " 1089   5.00e-02   1.49e-03   2.00e-02   1.00e-02\n",
      " 4225   2.00e-02   3.03e-02   3.00e-02   3.00e-02\n",
      "16641   8.00e-02   7.97e-02   6.00e-02   5.00e-02\n",
      "66049   4.00e-01   2.65e-01   3.00e-01   2.60e-01\n",
      "\n",
      "\n"
     ]
    },
    {
     "data": {
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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "option.plotflag = 0;\n",
    "pde = sincosNeumanndata;\n",
    "pde = sincosdata;\n",
    "mesh.bdFlag = setboundary(node,elem,'Neumann');\n",
    "femPoisson(mesh,pde,option);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Robin boundary condition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Multigrid V-cycle Preconditioner with Conjugate Gradient Method\n",
      "#dof:     4225,  #nnz:    20865, smoothing: (1,1), iter: 10,   err = 1.99e-09,   time = 0.018 s\n",
      "Multigrid V-cycle Preconditioner with Conjugate Gradient Method\n",
      "#dof:    16641,  #nnz:    82689, smoothing: (1,1), iter: 10,   err = 2.33e-09,   time = 0.062 s\n",
      "Multigrid V-cycle Preconditioner with Conjugate Gradient Method\n",
      "#dof:    66049,  #nnz:   329217, smoothing: (1,1), iter: 10,   err = 3.17e-09,   time = 0.21 s\n",
      "\n",
      " #Dof       h       ||u-u_h||    ||Du-Du_h||   ||DuI-Du_h|| ||uI-u_h||_{max}\n",
      "\n",
      " 1089   3.12e-02   4.92975e-03   4.34581e-01   2.56571e-02   8.30859e-03\n",
      " 4225   1.56e-02   1.24034e-03   2.17889e-01   6.44198e-03   2.08620e-03\n",
      "16641   7.81e-03   3.10581e-04   1.09020e-01   1.61223e-03   5.22032e-04\n",
      "66049   3.91e-03   7.76764e-05   5.45192e-02   4.03168e-04   1.30532e-04\n",
      "\n",
      " #Dof   Assemble     Solve      Error      Mesh    \n",
      "\n",
      " 1089   5.00e-02   1.39e-03   0.00e+00   1.00e-02\n",
      " 4225   2.00e-02   1.76e-02   2.00e-02   3.00e-02\n",
      "16641   8.00e-02   6.17e-02   6.00e-02   5.00e-02\n",
      "66049   4.90e-01   2.13e-01   2.50e-01   2.40e-01\n",
      "\n",
      "\n"
     ]
    },
    {
     "data": {
      "image/jpeg": 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     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "option.plotflag = 0;\n",
    "pde = sincosRobindata;\n",
    "mesh.bdFlag = setboundary(node,elem,'Robin');\n",
    "femPoisson(mesh,pde,option);"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Conclusion\n",
    "\n",
    "The optimal rate of convergence of the H1-norm (1st order) and L2-norm\n",
    "(2nd order) is observed. The 2nd order convergent rate between two\n",
    "discrete functions $\\|\\nabla (u_I - u_h)\\|$ is known as superconvergence.\n",
    "\n",
    "MGCG converges uniformly in all cases."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Matlab",
   "language": "matlab",
   "name": "matlab"
  },
  "language_info": {
   "codemirror_mode": "octave",
   "file_extension": ".m",
   "help_links": [
    {
     "text": "MetaKernel Magics",
     "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
    }
   ],
   "mimetype": "text/x-matlab",
   "name": "matlab",
   "version": "0.14.3"
  },
  "toc": {
   "colors": {
    "hover_highlight": "#DAA520",
    "navigate_num": "#000000",
    "navigate_text": "#333333",
    "running_highlight": "#FF0000",
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   "toc_section_display": "block",
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 },
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}
